Hyperbolic scale rule



May 4, 1937. H. oKURA l HYPERBOLIC SCALE RULE Filed March 20, 1933 l..ww .W L w f mm; mm.

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ma r1 a I t E R E 1 H -111 =am A llllllfllfll UNITED STATES PATENTOFFICE HYPERBOLIC SCALE RULE Ilisashil Okura, Tokyo, Japan, assignor toThe Hemm Seisakusho Co., Tokyo, Japan Application March 20, 1933, SerialNo. 661,743 In Japan January 14, 1932 2 Claims. (Cl. 235-70) Thisinvention relates to a slide rule and parthe scale L. C is thelogarithmic scale containing ticularly to a hyperbolic slide rule. Inany slide only one unit in the whole length of the scale L. rule nowknown, the scales for hyperbolic fune- CI is calibrated in a logarithmicscale similar to tions are invariably to work in compliance with C, butin the reverse direction thereto,

5 logarithmic scales. In such old slide rule, it is There are threescales (9), (R0) and (P) on the 5 required to provide three logarithmichyperbolic front face of the fixed bar J, shown in Fig. 1, scales forreading three hyperbolic functions, that which are, respectively, asfollows:- is, a logarithmic hyperbolic sine scale for hyper- (9), anon-logarithmic scale of angles, the bolic sine, a logarithmichyperbolic cosine scale angle being in degrees and decimals, and the l0for hyperbolic cosine and a logarithmic hyperwhole scale is for therange of O O-90. This scale 10 bolic tangent scale for hyperbolictangent. In is calibrated by the equation the log-log Vector slide rule,for instance, there o are provided a logarithmic hyperbolic sine scale 9:Smll/x and a logarithme hyperbolic tangent Scale, the along the scaleL, where x is the reading of the value of hyperbolic secant beingcalculated from corresponding point on the scale L. In the del5 thereadings 0f the hyPerbOlC tangent and the scriptlon hereinbelow has thesame meaning hyperbolic sine. as above mentioned. (R0), anon-logarithmic The prineipalobieet of this invention is to prof scaleof angle, the angle being in the circular vide a Slide rule having aSingle additional Scale measure or in radian, and the whole scale is forwhereby all six hyperbolic functions may be read the range 0f 20 on theslide rule by virtue of the single additional ,r scale. Other objects ofthe invention will appear 0-0 "2 f h h t th s ication l andglirgto mme tmug ou e Dec This scale is calibrated along the scale L by the Thisinvention is illustrated more or less diaequauon 25 grammatically in theaccompanying drawing, 9=sm1` Whereinin radian.

Flgm'e 1 1,5 a front plan Yiew of the scale; (P), a non-logarithmicscale, this is, a square Figure 2 1S a rear Plan ew of the scala rootscale, which is for the range of 0.0-1.0. 'I'he Like parts are indicatedby like characters equation dening the scale P is 30 throughout thespecification and drawing.

In the slide rule there are fixed bars J and H p=1/;

Secured together on both ends by plajtes K, with Where p is the readingon the scale P. This scale adequate Space between them' and m Whlch a isalso graduated taking the scale L as its sliding bar N is insertedbetween the fixed bars standard. 35 so as to slide longitudinally alongthe fixed bars. There are three scales (Q) I (Q.) and (C) on The threescales on the back ff the nxed bar the front face of the sliding bar, N,shown in Fig.

designated L, K, and A respectively, as shown 1n 1 which arerespectively as follows: Figure 2, are the same scales as those used on(Q) is exactly the same as (P).

40 ordinary polyphase slide rules. The three scales (Q'f' is anextension of (Q), from 1 0 to 1 414 40 on the back of the sliding bar Ndesignated which is the square root of 2 B' CI and C are also the saineScales as m95? (C), the common scale on the ordinary slide used onordinary polyphase sllde rules. That ls, rule a logarithmic scale of oneunit L, the equidivision scale, has the whole length Til th l (LL.)LI-"2) d (ma) divided declmally into 500 equal parts so that the ere areree sca es i an least one space is one five-hundredth part of the 01:1the front face 0f the xed bar H Shown in whole length of the scale. K isthe logarithmic Flg- 1 Which are the Same Scales aS the 10g-10g lscalecontaining three units in the whole length Scales that are fOurld 0h the10g-'10g duplex Slide of the scale L, and A and B are the logarithmicrule.

scales containing two units in the whole length of There are threescales on the back of the xed 50 bar J designated (D), (T), (G0) whichare the scales mentioned below,

(D), equal with said (C) scale;

(T), a non-logarithmic tangent scale, graduated from 0.0 to infinity, bythe equation t=tan 9, where e is the reading of the point on the scaleR0 corresponding to the reading t on the scale T. This scale is derivedfrom the scale R6.

(G0), is the Gudermanian scale which characterizes the presentinvention. This scale is graduated by the equation u tanh-H/ f oru=log(tan e+sec e) where u and e represent the readings of thecorresponding points on the scales G6 and R0 respectively, theGudermanian scale being derived from the scale L or R6.

The slide rule of this invention is of the duplex type. It has eighteenscales in all on it. Among these scales, the following seven scales areparticularly important: the equidivision scale L, the square root scalesP, Q and Q', the radian scale R0, the tangent scale T and theGudermanian 'scale G0.

Now, I will explain the particularadvantages of this slide rule. Put thehair line on u on the scale G6 and let x be the reading on the scale Lcoming under the hair line. Then :c will be tanh2 u, becausex=sin2e=sin2(gdu), and sin (gdu): tanh u by the Gudermanian theorem.

Examples.-Against u=0.5, u=1.0, u=2.0 etc. on (G0) there arerespectively tanh2 (0.5) 0.2135, tanh2(1.0)=0.5800, tanh2(2.0)=0.9305,etc. on (L). The scale of this invention is thus entirely different fromany scale on the Mannheim slide rule; I will explain the relationshipbetween the various scales on the present slide rule mathematically. Asabove mentioned, u=log (tan e-l-sec. e).

Now from this formula can be derived sin 9+ 1 cos@ where e is the baseof Napierian logarithms.

Hence sin 9= Hence sin 9=tanh u which relation is the most essentialbasis for this invention. Now let e: gd u, then sin (gdu.) =tanh u Ifthe hairline be set at :15, on (L), and read 9, u, p and t respectivelyon (R0), (G0), (P) and (T) under the hairline; then the result is:

Thus, if the hairline be set at a reading p on (P), then p itself istanh u or the hyperbolic tangent of u, where u is the very reading on(G0) under the hairline.

Again as has already beer. stated, the value of p is the sine of 9, orpsin 9, 0 being the reading on (R0) under the hairline; and 9 on (Rl) ilequal to the Gudermanian of u, or 9=qdu, u being the reading on (G0)under the hairline. But by Now by one of the well-known formulae forhyperbolic functions:-

cosh u-sinh?u=1 So the value of t on (T) found for u on (G0). is equalto the value of the hyperbolic sine of u, or to sinh u, Next, if theleft end of (Q) beset to u on (G6), and the reading of the point on (Q)against the right end of (P) is y, then by the natures of (P) and (Q) weshall have Hence lSo it will be seen that one can have sech u on (Q)against the right end of (P) when the left end of (Q) is setto u on(G0). Also by the formulae of hyperbolic functions:-

colech u= smh u mh "m Thus it is clear that when one has sinh u, tanh u,and sech u, he could very easily have the -other three functions cosechu, coth u and oooh u by simply taking the reciprocals of the former on(CI).

Thus one can have all the aix classes of hyperbolic functions by theaddition of a single scale (G0) to an existing slide rule.

EXAMPLES 0F CAITULATION (a) sinh u.

Set the hairline at the given value of u on (G0) and one can read sinh uon (T) under the hairline.

Example 1.-sinh 0.32-:0.325.

Put the hairline at 0.32 on (G0) and read 0.325 on (T) under thehairline.

(b) tanh u Set the hairline at the given value of u on (G0) and one canread tanh u on (P) under the hairline.

Example 2.-tanh 0.8320581.

Put the hairline at 0.83 on (G0) and read 0.681 on (P) under thehairline.

(c) sech, u

By the help of the hairline, set the left end or 0 of (Q) to the givenvalue o1' u on (G0) and (d) cosh u Reverse sech u by the help of thehairline on (CI) Example 4.--cosh 0.55=1.154.

Put the hairline at 0.868 on (C) and read 1.154 on (CI) under thehairline.

(e) com u Reverse tanh u by the he1p of the halrune on (CI).

Example 5.-coth 0.83=1.468. Put the hairline at 0.681 on (C) and read1.468 on (CI) under the hairline.

(f) cosech u ybar having a square root scale bars secured together onboth ends with space therebetween and a bar slidably inserted into thespace between the said xed bars, one ilxed bar having a square rootscale on its front face and a Gudermanian scale, a tangent scale and alogarithmic scale on its back face, and the sliding and a logarithmicscale on its front tace and a reversed logarithmic scale on its backface, and the other flxed bar having an equidivision scale on its backface, all of the said scales being calibrated along the length of thebar taking the equidivision scale as the standard scale.

2. A slide rule comprising two xed parallel bars secured together withspace therebetween and a sliding bar slidably inserted into the spacebetween the said ixed bars, one xed bar having an angle scale in degree,a radlan angle scale and a square root scale on its front face, and aone unit logarithmic scale, a tangent scale and a Gudermanian scale onits back face, and the sliding bar having two square root scales and aone unit logarithmic scale on its front face and a two unit logarithmicscale, a reversed logarithmic scale and a one unit logarithmic scale onits back face, thel second fixed bar having three log-log scales on itsfront face and an equidivision scale, a three unit logarithmic scale, asquare scale and a two unit logarithmic scale on its back face, all ofthe vsaid scales being calibrated along the length of the bar taking theequldivision scale as the standard scale.

H. oKUaA.

